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Type Decomposition and the Rectangular AFD Property for W*-TRO’s
Published online by Cambridge University Press: 20 November 2018
Abstract
We study the type decomposition and the rectangular $\text{AFD}$ property for
${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$. Like von Neumann algebras, every
${{W}^{*}}-\text{TRO}$ can be uniquely decomposed into the direct sum of
${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of
$\text{type}\,I,\,\text{type}\,II$, and
$\text{type}\,III$. We may further consider
${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of
$\text{type}\,{{I}_{m,n}}$ with cardinal numbers
$m$ and
$n$, and consider
${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$ of
$type\,I{{I}_{\lambda ,\mu }}\,\text{with}\,\lambda ,\,\mu \,=\,1\,\text{or}\,\infty $. It is shown that every separable stable
${{W}^{*}}-\text{TRO}$ (which includes
$\text{type}\,{{I}_{\infty ,\infty }}$,
$\text{type}\,I{{I}_{\infty ,\infty }}$ and
$\text{type}\,III$) is
$\text{TRO}$-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for
${{W}^{*}}-\text{TRO }\!\!'\!\!\text{ s}$. One of our major results is to show that a separable
${{W}^{*}}-\text{TRO}$ is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular
$\mathcal{O}{{\mathcal{L}}_{1,{{1}^{+}}}}$ space (equivalently, a rectangular
$\mathcal{O}{{\mathcal{L}}_{1,{{1}^{+}}}}$ space).
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- Copyright © Canadian Mathematical Society 2004
References
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